Is spacetime simply connected?

I suppose there are many aspects to look at this from, anna v mentioned how Calabi-Yao manifolds in string theory (might?) have lots of holes, I'll approach the question from a purely General Relativity perspective as far as global topology.

Solutions in the Einstein Equations themselves do not reveal anything about global topology except in very specific cases (most notably in 2 (spacial dimensions) + 1 (time dimension) where the theory becomes completely topological). A metric by itself doesn't necessarily place limits on the topology of a manifold.

Beyond this, there is one theorem of general relativity, called the Topological Censorship Hypothesis that essentially states that any topological deviation from simply connected will quickly collapse, resulting in a simply connected surface. This work assumes an asymptotically flat space-time, which is generally the accepted model (as shown by supernova redshift research and things of that nature).

Another aspect of this question is the universe is usually considered homogenous and isotropic in all directions, topological defects would mean this wouldn't be true. Although that really isn't a convincing answer per say...

Benjamin Horowitz's answer covered a lot of the key points, but it's worth adding that the question of the topology of the universe has been investigated by astrophysical observations. If the universe is multiply connected, and if the length scale is shorter than the horizon scale, then we should be able to see evidence of it.

To take a simple example, imagine that the universe is geometrically flat but has the geometry of a 3-torus. Specifically, take a cubical volume, and identify opposite faces, so that if you "leave" the cube through one face you reenter through the opposite face. If the length of an edge of the cube is sufficiently small, then you could see multiple copies of any given object. Of course, if the length is much larger than the horizon, then there's no way to tell the difference between this model and one in which space is infinite.

The best way to test these models is the "circles in the sky" technique, in which you look for correlated circles in different directions in maps of the microwave background radiation. The result is negative: we don't live in a multiply-connected universe with a sufficiently short length scale to be observable.